1 Antwort

Difficult to answer if you don't specify a particular path of the structure, since we have to compare here two path formulas with each other. Let me therefore draw your attention to the following path that is included in the above structure:

Recall that [x SB y] holds on a path at some time t, if and only if at some time x holds and up to that point of time (including that point of time) y does not hold on that path.

Indeed, the only initial path of the above structure can distinguish between the two formulas:

[b SB c] holds at no point of time on that path since b never becomes true on the path

[a SB [b SB c]] holds therefore at all points of time, since a holds from t=1 on, and [b SB c] never holds on the path, thus a holds before [b SB c]

[a SB b] holds at all points of time on that path since b never becomes true, but a does from time t=1 on, so a is true before b

[[a SB b] SB c] does not hold at time t=0 on the path since c is already true there and therefore [a SB b] cannot become true before c

To summarize, on that initial path, [a SB [b SB c]] holds at time t=0, but [[a SB b] SB c] does not hold there. Hence, the structure distinguishes between the two formulas.

If you want to double-check the above claims with a tool, then you can check the following formulas for validity: